The probability of recovering is exactly half of what you estimate it to be due to the placebo effect/positive thinking.

It would take an artificially bad situation for this to be the case. In the real world, the placebo effect still works, even if you know it's a placebo--although with diminished efficacy.

But that's beside the point. More on-point is that intentional self-delusion, if possible, is at best a crapshoot. It's not systematic; it relies on luck, and it's prone to Martingale-type failures.

The HPMOR and placebo examples appear, to me, to share another confounding factor: The active ingredient isn't exactly belief. It's confidence, or affect, or some other mental condition closely associated with belief. If it weren't, there'd be no way Harry could monitor his level of belief that the dementors would do what he wanted them to, while simultaneously trying to increase it. Anecdotally, my own attempts at inducing placebo effects feel similar.